A thermoplastic elastomer composition obtained by, so-called dynamic crosslinking, which is crosslinking during melt kneading of a radically non-crosslinkable resin such as polypropylene (PP) and a radically crosslinkable elastomer in an extruder in the presence of a radical initiator, is widely used in machine parts and the like. As such rubbery compositions, an olefin-based elastomer composition or the like using ethylene-propylene-diene rubber (EPDM) is known (see Patent Documents 1 and 2). A composition obtained by dynamically crosslinking hydrogenated rubber is also known (see Patent Documents 3 and 4). Furthermore, disclosed is a thermoplastic elastomer composition containing an olefin-based resin, and a hydrogenated product of a copolymer of a vinyl aromatic compound and a conjugated diene compound (see Patent Documents 5 and 6).    Patent Document 1: Japanese Patent Laid-Open No. 8-120127    Patent Document 2: Japanese Patent Laid-Open No. 9-137001    Patent Document 3: Japanese Patent No. 2737251    Patent Document 4: Japanese Patent Laid-Open No. 2004-67798    Patent Document 5: Japanese Patent Laid-Open No. 2001-49051    Patent Document 6: Japanese Patent Laid-Open No. 2005-89656